Surface bundles and the section conjecture

We formulate a tropical analogue of Grothendieck’s section conjecture: that for every stable graph Γ of genus g> 2 , and every field k, the generic curve with reduction type Γ over k satisfies the section conjecture. We prove many cases of this conjecture. In so doing we show the existence of many examples of curves with no rational points satisfying the section conjecture over fields of geometric interest, and then over p-adic fields and number fields via a Chebotarev argument. We construct two Galois cohomology classes o₁ and o₂~ , which obstruct the existence of π₁-sections and hence of rational points. The first is an abelian obstruction, closely related to the period of a curve and to a cohomology class on the moduli space of curves M_g studied by Morita. The second is a 2-nilpotent obstruction and appears to be new. We study the degeneration of these classes via topological techniques, and we produce examples of surface bundles over surfaces where these classes obstruct sections. We then use these constructions to show the existence of curves over p-adic fields and number fields where each class obstructs π₁-sections and hence rational points. Among our geometric results are a new proof of the section conjecture for the generic curve of genus g≥ 3 , and a proof of the section conjecture for the generic curve of even genus with a rational divisor class of degree one (where the obstruction to the existence of a section is genuinely non-abelian).